Fekete points

Remarks

A system of Fekete points exists for each $n$, since $K^n$ is compact and the product is continuous. It need not be unique: a rotation of Fekete points of a circle centred at $0$ gives Fekete points again. The points are distinct, because the maximum is positive. The maximum equals $\delta_n(K)^{n(n-1)/2}$, where $\delta_n(K)$ is the $n$-th diameter in the definition of the transfinite diameter.

Example: on the unit circle, the $n$-th roots of unity $1, \omega, \dots, \omega^{n-1}$ with $\omega = e^{2\pi i/n}$ are Fekete points. The product of their mutual distances is the absolute value of their Vandermonde determinant, which equals $n^{n/2}$. Hadamard's inequality bounds the Vandermonde determinant of any $n$ points of the closed unit disc by $n^{n/2}$, so the maximum is attained.

Non-example: for $n = 2$ on the unit circle, the points $1$ and $i$ are not Fekete points, since $\abs{1 - i} = \sqrt 2 < 2 = \abs{1 - (-1)}$.

Maximizing the product is the same as minimizing the discrete logarithmic energy $\sum_{i < j} \log(1/\abs{z_i - z_j})$.